Kinetic theory of microphase separation in block copolymers

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Kinetic theory of microphase separation in block copolymers

E.L. Aero, S.A. Vakulenko, A.D. Vilesov

To cite this version:

E.L. Aero, S.A. Vakulenko, A.D. Vilesov. Kinetic theory of microphase separation in block copoly-

mers. Journal de Physique, 1990, 51 (19), pp.2205-2226. �10.1051/jphys:0199000510190220500�. �jpa-

00212522�

Kinetic theory ^of microphase separation ^{in block} copolymers

E. L. Aero, S. A. Vakulenko and A. D. Vilesov

Institute of Macromolecular Compounds, ^USSR Academy ^of Sciences, V.O., ^Bolshoi pr. 31,

199004 Leningrad, ^U.S.S.R.

(Received ^on February 21, 1990, accepted ^on May ^22, 1990)

Abstract.

A semiphenomenological theory that has the advantage ^of taking ^{into account}

nonlinear and nonlocal contributions in the free energy for microphase separation ^{of block} copolymers ^is proposed. ^A kinetic nonlinear equation defining ^the process of structure formation from a melt is obtained, ^{and its} analytical ^{solution at} the melt-structure transition temperature ^is examined. In this region, ^{the structure formation} proceeds ^{in two} stages. ^{The first one is} characterized by damping of all but stable Fourier-components ^of density distribution and the second, by stabilization of the amplitude ^{of the} distribution. Characteristic times of these processes ^are estimated. The applied approach ^{allows a} comparatively simple definition of lamellar, hexagonal ^and body-centered ^cubic structures near Ts. Equilibrium structures at T ~ Ts ^{are described as well.}

Classification

Physics ^Abstracts

61.40K

64.75

68.45

1. Introduction.

It is known that the onset of the incompatibility ^of copolymer constituents at a certain temperature lower than T, ^{leads to} the formation of microdomain structures. The domains consist of segregated components ^{of block} copolymers ^{and are} periodically spaced. ^The morphology ^of structures is determined by ^the composition of the block copolymer. ^{For block} copolymers with molecular mass of the sequence appreciably smaller than the other, ^{a cubic-} body ^{centered structure is formed} by spherical domains of the minor component. ^{When the} minor component ^{is in} larger ^amount the domains represent cylinders packed ⁱⁿ hexagonal lattice, ^{and when} component ^{fractions are} nearly ^{same in} magnitude, ^{lamellar structures}

appear.

For the last two decades the interest on the morphology ^{of block} copolymers ^{has been} extremely high ^and considerably large ^{number of} experimental and theoretical investigations [1] have been dedicated to them. The reason of this interest seems to be the fact that block

copolymers ^{are the} simple ^{model of} self-organization, ^{and that} they allows the regulation ^of supermolecular ^structure properties ^{at the} stage ^{of block} copolymer synthesis. They ^{are also} convenient objects ^{to model} heterogeneous systems ⁱⁿ which the dimensions and the

morphology ^of inhomogeneities ^can be controlled with a relative precision.

Theoretical studies dedicated to the thermodynamic properties ^{of a} microdomain structure may be divided into two main _{groups :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510190220500

1) ^studies dealing ^{with an} equilibrium ^structure of the block copolymers ^{at T «} Ts. ^{It is} supposed ^{that for a} system ⁱⁿ equilibrium, ^the morphology ^is pre-assigned, microphases ^are properly ^{divided and the} interlayers between them are narrow. The task undertaken is to determine the relationship of structural parameters ^{with the molecular} characteristics of block

copolymer [2-7] ;

2) ^studies dealing ^with equilibrium properties ^near T,, ^where appearing ^{structure have}

wide interlayers between the domains. These theories are based on a Landau-type representation of the free energy [8-10].

The theory described below presents ^the equilibrium and stable states of structures, with T - Ts ^{as well as with T} Ts, within the limits of a single model, and allows the formulation of the kinetic equation ^{of the} process of structure formation from ^a melt, ^{and to} investigate ^its

solution. The importance ^{of the account of} nonlinear effects is proved. ^{It is shown} that, ^when describing ^the equilibrium ^{states near} T., ^the approached theory ^{is a} simplified ^{version of}

Leibler’s theory [8]. However, ^the theory describes the same structure of phase diagrams ^and

the ^same dependence ^{of structure} parameters ^on the number of segments ^{in the} chain, ^{on the}

fraction of components p o and _x. The mathematical technique ^{used in our} theory also allows

us to describe the structure at T « Ts. ^The dependence of the free _energy ^on the molecular characteristics obtained in this case agrees with the dependence described in previous ^studies [5-7].

Thus, ^the importance ^{of the} suggested approach ^{is in the} possibility ^to investigate ^{the stable}

states and the kinetics of structure formation at T - T,, ^{and stable states at T} 7g ^{on the basis}

of a single ^model.

In the present article the kinetics near 7g ^has been examined. The Kuramoto-Tsuzuki method used by ^us [ 11 allows ^{us to} get ^the equations ^{of structure} formation that have been examined both analytically ^and numerically. ^The theory also allows ^us to calculate the time of relaxation. It should be noted that the present ^kinetic approach ^{in the} region ^near 7g ^can be carried over to a more precise ^model [8]. Actually, ^{to formulate a kinetic} theory, ^{it is} quite sufficient to have a free _energy functional and a dissipation ^function. Onsager’s coefficient, being ^the expression ^{for the} dissipation function, ^was used in the form obtained for the _process of segregation ^of polymers ^{in a} binary ^mixture [12].

2. Construction of a functional of free energy.

To obtain the expression for the functional of the free _energy for the melt at T s T,, ^{it is}

necessary ^to express ^two effects : 1) ^local (segregation ^of segments ^{A and B in a} system ^of unconnected blocks) ^and 2) ^non-local (the ^increased entropic elasticity that appears during

the segregation ^{in a block} copolymer macromolecule, and that limits the fluctuation value of

density ^of segments ^{A or} B, i.e., ^domain’s dimensions). The first effect can be described by

means of a lattice functional [13]

where _p

_p (x) ^{is a local} density ^{of number of} type ^A segments ^at point x, X is Flory parameter, a ^{is the} length ^{of a} segment, NA, NB ^{are the numbers of} segments ^{A and B in a} molecule : NA ⁺ NB

^N.

Concerning ^{the second} effect, it should be described, owing ^{to its} non-locality, by ^{the terms}

of a form

f K(x _ X,) P (X) P (X,) d3X, ^where A"(x-x’) ^{is a certain} function, integrally

dependent ^{on p} (x). ^The presence of such terms in the functional of the free energy hampers

the analytical investigations seriously. Therefore, ^the simplest approximation ^{of this term is}

found. When performing the Fourier transform, ^we get ^a part ^{of free} energy, square-law ^with respect ^{to p}

where S- 1 (K ) ^{is an inverse} scattering function, ^{K is a wave vector.}

Calculations [8] show that S- l(K) ^{has a} singularity, ^when K - 0, in the form of

a 1 K 1- 2. ^{It appears that the} longwave fluctuations of the density ^{of A and B} segments ^result in an intensive growth of the free _energy, and, therefore, ^are unfavorable. Thus the

expression (2.2) really ^describes the limitations of regions ^of segregation. ^It is convenient to describe this effect in _x-space by ^{means of a vector order} parameter

The condition (2.4) is necessary for the restoration of ^P with respect to p. It should be noted

f 3X fallows rot ^{NA -}

that from the condition of minimalit y

of L . v ^{P2 d3x follows rot P}

^{0. In (2.3), po}

NA N N ^{is an} average density of the number of the segments ^{A at a} point ^{x, V} is the volume of the system.

It should be observed that an analogous example ^was used in reference [14]. Then, ^the

contribution of the singularity ^{mentioned above} a 1 K 1- 2 ^{in the} expression (2.2) ^{can be}

rewritten in the form

When differentiating (2.5) ^{we used the} relationship between the Fourier components following ^from (2.4)

a-coefficient can be calculatéd from the asymptotic of the function S(K ) [8], ^when

that can be also shown by ^{means of} calculating ^the changes ^of entropy ^{of block} copolymer

molecules at segregation [10]. So, ^the simplest way ^to take the limitation of large-scale

fluctuations into account consists in entering ^{the term a P2 in the} density functional of the free energy.

Thus a complete expression for the free energy density ^{in our model is}

The free energy functional :F contains the square of second P-derivatives with respect ^to

space parameters. The local functional mentioned above is a simplified ^one. However, precise

mathematical results [15] show that further simplification ^is impossible. Actually, ^{if we}

eliminate the term with a P2, the emergence of structures (conditions ^{with P fl} 0) ^is possible,

but these will be unstable. In our model, ^{as will be shown} below, structures are stable. The functional :F from (2.8) ^{can be described as the} simplest ^model approximation ^{to the}

functional examined in [8].

The relation between this theory ^{and our} model in the vicinity ^of T, ^{is described later in}

detail.

3. Kinetic équation ^for the evolution.

In our model, the formation of structure from a melt is characterised by the emergence of ^a

non-zero parameter ^{P. The} equation describing its evolution can be written in the form [ 11 ]

where D is functional assigning ^a dissipation ^{in the} system. ^The sign ^{in the} equation (3. 1) ^is

determined by the condition

which means that, when the time t is growing, ^{the free} energy does not increase.

The positive functional D has been chosen in the form

The form D and functions 7y(p) will be discussed below.

It should be noted that the condition (3.2) ^{follows from} (3.1). Multiplying both sides by P, ^{and then} integrating ^with respect ^{to the whole} volume, ^{one obtains}

The equality ⁱⁿ (3.4) ^can be obtained only ^when Pt - 0, i.e., ^{in a} stationary ^{state. Since the}

functional does not increase, and, ^as will be shown later, it is limited from below (uniformly

with respect ^to P(x)), ^the following limits exist ^{t - 00} lim Y (t), t - 00 ^lim dt == dF ^{0. But according to (3.4)} the equality dy 2013 = ^{0 is} equivalent ^to the condition D

0. Thus, ^{one can not} only ^{show the}

dt

existence and the limitation of solutions but also affirm that all solutions (3.4) ^{tend to} equilibrium for which Pt ^{= 0. The} general mathematical theory ^of Cauchy initial value

problem ^as such is well developed [16]. ^The property (3.4), ^{in the bounds of a common} approach, allows the description ^of either the kinetics or the stable states, since ^the

equilibrium ^{states are the limits} (when t ^--+ oo) solutions of Cauchy ^{initial value} problem. ^It

should be noted that the form of stable states does not depend ^{on the} form of D-function

every choice gives ^{the same stable states. But it seems to us that the} investigation of kinetics allows a more profound understanding ^{of some} experimental ^facts pertaining ^{to the stable}

states that will be shown in section 8. The choice of D, e.g., in the form of (3.3) ^{has an effect}

on the time of relaxation.

It should be noted that equation (3.1) ^is equivalent, ^{when a}

^{0, to a diffusion} equation

used in a study [12] ^to describe the spinodal decomposition ^{of a} polymer binary mixture. The difference is in form of the free _{energy 37.}

Thus, in reference [12], ^the equation ^used (in ^our notation of variables) ^was

where

Taking ^{into account}

This is equivalent ^{to the} equations

when ^«

0. It should be observed that, when a :0 ⁰ these transformations show the

possibility ^to write the kinetic equation in the form (3.5) ^without entering ^order parameter ^P.

But, ^{in this} case, ^a nonlocal integral ^{term appears in Y.}

4. Investigation of the kinetic équation ^near TS ^{at first} stage of evolution.

Let us present the function (2.8) ^near Tg in the form of a series with respect ^{to P and its} derivatives

If (4.1) is taken into account, equation (3.1) ^will get ^{a rather} complicated ^{form. We will not}

write it, since it is worthless for our later investigations. Expressions for the coefficients (3, K,

8, y, K 1, 81 through X, ^{N, a,} p o have been obtained by ^means of matching up the terms of(4.1)

with the series in the vicinity ^{of p =} p o, taking ^{into account} NA

p o N, NB = ( 1 - p 0) N :

The term c div P does not give any input if it is included in equation (3.1). ^{Let us use the}

function in the form obtained in [12], where the kinetics of the spinodal decomposition ^{of a} polymer binary ^{mixture was} searched. For it seems to us that near T, ^{where the} segregation ^of

the block A and B is insignificant, ^the connectivity of the blocks A and B will not have a

serious effect ^on diffusional flows through ^{a chemical} potential gradient. ^The Onsager coefficient, ^{in our} case, ^as well as for a mixture, ^can be calculated by the formula

where Ne ^{is a} number of segments ^between two nearest catchings along ^{the chain,}

TA, TB ^are microscopic relaxation times for the segments ^{A and B.}

Thus

r- r,

It should be noted that the solution of the equation, ^when T, S ^T ) ^{1, depends on the}

choice of q (p ), ^{but to an} inconsiderable degree. Actually, within the bounds of the

approximation ^taken

since the deviation of p from p o of E order, ^{as we} will show later. The account of the term of e order should only ^{lead to the} change ^{of 0 (--)} in the relaxation time. But the behaviour of solutions and equilibrium ^{states do not} change after that. In the _process of simplification ^the

left side of (3.1) ^{assumes the} form q 0 P,, and the whole equation, ^{the form}

To study (4.6), ^{let us state the} Cauchy initial value problem. Experimental facts show that the formation of the structure and its properties, particularly ^a space period, ^{do not} depend

on the size of the system ² C, ^{when C is} large enough, and it is not necessary to account for

boundary ^{effects. Let us choose, as} boundary conditions for (4.6) that allow the elimination of the boundary effects, ^the boundary conditions of periodicity

Putting additional conditions on initial data

The condition (4.8) ^with (2.3) unambiguously determines Po ^with respect ^to p o.

If t

0, ^{rot P}

0, then, when all t > 0, ^{we’ll see the same results. This can be} proved by calculating ^rot from both sides of (4.6). ^{And for t} > 0 the condition f v ^{v P d 3X}

^{0 is preserved}

analogously. ^{Let us examine some} properties ^{of the} solutions of the initial value problems (4.6) ^{for P.} First, ^we show that Y is bounded from below. The coefficient ^a is positive, f o according ^to (2.1 ) is bounded below and for these reasons, the functional is always ^bound

below for ^a space-bounded system. That is also true for ^a simplified notion, ^since a, 8, ^K,

y 1 are positive, ^{and the} corresponding ^terms predominate.

Thus, the theoretical observations cited in section 3 are obviously ^true. Any solution of the

Cauchy initial data problem ^{aims at the} equilibrium solution. This equilibrium ^solution

satisfies Lagrange-Eiler’s equation

The physical meaning ^{of these} mathematical statements is that (4.6) ^is really ^{a kinetic} equation describing ^the approach ^{of the} system ^to the stable state. Now we should investigate,

when the stable state matches the structure ( t ^-+ oo ) ^{and not a} melt, ^i.e.

Let us find a critical value of the parameter /3, ^{so that the} solution, ^{even with an} arbitrary

initial amplitude, ^{does not} degenerate (when t ^--+ oo) into trivial (P == 0). ^{We’ll consider} the initial amplitude ^to be small. Then it is possible ^{to linearize (at least in some initial} part) ^with respect ^{to 0 -- t --} r 1. Representing by ^{Fourier series, we} get

obtained from (4.6) for Fourier components PK

Rejecting ^{the term} 0 (Pi), ^{we write} (4.11) in the form

The solutions PK ^are always damping, ^{if a,} /3, K > 0. Let us find the critical values

,8 (X ) ^and K, ^{when the} equilibrium ^structure originates from the conditions

From (4.13) ^{we obtain}

As a result of the substitution of (4.2) ⁱⁿ (4.14) ^we obtain the explicit value of the expression

for critical _XS from (4.14), ^{and the structure} appears

Thus, ^when p o

0.5, the critical value NX s e- 9.6 ; ^when p o

0.3, NX, -- 13.9. The values

NX calculated for these values _{p o} are 9 and 14 in Leibler’s more precise theory. ^The period ^of

the structure is described as

As a small parameter ^{for the} analysis of the solutions (3.1), ^{it is} quite ^{natural to choose a value}

E = J 1 X - X s 1 - J f3 2 - 4 a K ^that determines the presence of ^roots in the function

g(K). ^The negative g(K) correspond ^to non-damping ^{modes K. It} is obvious that the relaxation time, ^{in this} case, will be - ê - 2. The relationship (4.14) determines the sphere ⁱⁿ

the _space of wave vectors that, ^{in the case of} non-damping ^{modes are} located in spherical layer ^{with a} depth ^{of order E around the} sphere. Harmonics with wave vectors K whose tips

are located at distances » E from the critical sphere ^are, according ^to (4.12) exponentially damping during times ...c E- 2

Thus the analysis of the linearized equation ^{carried out allows us} to extract the critical modes K, ^{and to} obtain critical value _X,. But one should not neglect the nonlinear terms at times of ê - 2 order. Besides, ^{nonlinear terms are} important ^when describing ^{a form of a stable}

state. The amplitudes ^{of the} density ^deviation p - p A - p o ^cannot be calculated within the limits of the linear approach, ^since the solution of a linear equation multiplied ^{into a constant}

is also its solution. Taking into consideration the facts about the modes, damping ^{and not} damping ^{at the first} stage ^of evolution, the Kuramoto-Tsuzuki approach ^{seems to be the most}

natural one. It allows, ^at the small quantities p, obtain from (3.1) ^a simplified ^kinetic equation. ^{To make the} procedure ^more simple, ^{let us} examine first the one-dimensional case.

5. Investigation ^{of kinetic} équation by ^means of the Kuramoto-Tsuzuki method.

The asymptotic solution for the system of reaction-diffusion type has been obtained in [17].

The solution satisfies a simplified equation ^{which has one and the same} form for all systems.

In this case we examine an analogical ^solution

where f/1 ^{is a new unknown} complex amplitude, p 1 is ^a correction to the asymptotic ^solution.

Thus we go ^{on to} the new « slow » parameters X, ^T. The choice X

ex is determined by

the fact that a value interval K, ^{for which} &(K) : ⁰ (and corresponding ^{modes do not} damp exponentially) has the dimension of ^e order. The choice of T

E2 t is connected with the fact that the speed ^{of mode} change ^{does not exceed} min g(K) = O ( E 2). The modes for which

[ K - K, » ^{0(e) prove to be} exponentially small after the end of the evolution’s first stage described in the previous section. For the ^reasons mentioned above it is quite clear that the solution of the equation should be found in the form (5.1 ). ^{In this} case, the « envelope » ^{in the} beginning of the second stage of the evolution (after ^the damping of all modes :

IK - Ksi ^{0 (e» when the solution is presented} in the form (5. 1), is determined from ^a

relationship

where _T is ^{some moment of time,} 7-1 «.,c 0 (£- 2). T 1 is ^{the time} of damping ^{of all} harmonics K,

for which s (K ) > 0, 1 g(K) [ > 0 (e 2) . ^The option ^of T is conventional to a great ^extent (in ^the

limits described), but this is not important ^{for our} subsequent research. The reasons

mentioned do not allow us to insist that the solution _{p as} is correct, ^{when t »} r 1. Actually,

because of the _presence of nonlinear terms in one-dimensional version (4.6) :

an apprehension ^arises that, ^{when t} > T 1, in the solution p ^{the terms} increasing ^with respect

to t and proportional ^to precedent ^harmonics e2 iKs x , e 3 ZxS x,

will arise.

In Kuramoto-Tsuzuki method f/J(X, T ) changes ^{so that the terms} connected with

einK, x. n - ² remain correctional to p as from (5.1), ^{when t --+ oo. The} approach [ 17] consists in the fact that (5.1) ^is represented ⁱⁿ (5.2), ^{and once} the calculation in the right hand side has been made, requires the correction _p to be limited as 0 (e 2)@ ^{with t - oo. This is} impossible

with the arbitrary ^values f/J. ^{But it turns out} that if the changing amplitude e satisfies the

equation

the correction _{p 1} remains limited.

As ^was shown in [17], ^the Kuramoto-Tsuzuki equation ^{is a} universal kinetic one for the

vicinity ^{of a critical} point ^{in a nonlinear} dissipative system. ^{One can} show that for (5.3) ^the

asymptotic satisfies :

only ^{if at} starting point f/1 ^{was not} equal ^{to zero.} According ^{to the} general theory [16] ^the asymptotic (5.4) is violated only ^{for «} exceptional » ^{initial date, and we} will discuss this below.

Before calculating the coefficients _{cl, C2, c3,} let us discuss the properties ^{and the} physical meaning ^of equation (5.3). ^{Enter a functional}

Multiplying ^{both sides} of (5.3) by f/I-r(X, T), ^{where a star means} complex integration, ^and integrating ^with respect ^to X, ^from (5.3) ^we obtain that in the solutions of this equation

where

Equation (5.3) ^{itself can} be written by ^means of the functionals f and D :

Hence it _appears that equation (5.3) ^{has the same} functional structure as the initial

equation (3.1). But it is much simpler and allows the investigation ^{in a} complete ^{form. The} general theory ^of equation ^as such says that ^« almost for all » initial date the solutions of the

Cauchy initial value problem ^for (5.3) ^{tend to} stable the extremes of the functional

Y. The results also show that a unique ^{stable extremum of F is the} solution of the form

for which Y has a global ^minimum equal ^{to zero.}

Let us review our considerations. The density ^{p is «} almost » harmonic at the second stage

of the kinetic evolution. The evolution comes to the increase and the equalization ^{of the}

amplitude 14r 1 that is the envelope ^{of this} harmonic, slowly transforming in space. The time ^T

required ^{for this} process, ^{as can} be shown, ^is approximately ^{evaluated as}

Formula (5.9) ^{is correct} when the small initial fluctuation that prescribes initial data in (4.6)

is much smaller than c with respect ^{to the} amplitude. ^{If this} condition is not observed, ^the

method cannot be applied ^{at all. This} analysis (Sects. 4, 5, ^with applying the modern theory ^of

non-linear equations) proves that for all sufficiently small initial data at T T,, the evolution with respect ^{to the kinetic} equation (5.7) ^{comes to a} space-periodical ^{solution p =}

s - sin C2 . _{(K, x).} ^In the limits of the system’s large dimensions (£ - oo ) the solution period

C3

does not depend ^{on L} It should be noted that this analysis ^{is for a} finite dimension system ^as well (£ - E - ’). ^{And if} £ « E - 1, ^{it means that the} term CI f/J xx should be eliminated from the Kuramoto-Tsuzuki equation.

At the first stage ^{of the} evolution, when all harmonic’s amplitudes « ^{E, their} damping ^time (excluding ^{those for which K E} (K, - E, K, ⁺ 8» ^is Tl’" e-2 710’ ^Thus the relaxation time of the enveloping amplitude 4r ^{to a constant} (according ^to (5.9)) ^is In JAI times larger ^{than the}

C2

harmonic’s damping ^{time. In the next section we} will show the method of calculating ^the

constants cl, c2, c3, that determine ^a stable amplitude ^{and a} relaxation time, and also the method of deriving (5.3) ^from (4.6) ^and (5.1).

6. Calculation of cl, c2, c3, in one-dimensional case.

The Kuramoto-Tsuzuki method described in section 5, ^if appplied ^to (4.6) ^{in its} explicit ^form,

leads to awkward calculations. These calculations can be simplified, ^{if we address Whitham’s}

idea [18] ^{that was} applied ^to the non-linear equations’ theory long ago. Actually, ^{for the} Lagrange-Eiler equation which describes stable states

the following approach ^{can be} brought forward. Let some asymptotic ^formula giving ^an approximate ^solution be found. This formula usually ^contains slowly changing parameters.

E.g., ^{in our case}

Then we assert that the equation ^{for slow} parameters ^{can be found} by calculating ^{the free}

energy :F:)

:F[Pas] ^and by varying ^{it with} respect ^{to the slow} parameters

This idea can be generalized ^{to the kinetic} equation ^{of the} (3.1) ^{form. It is} enough ^to

assume that the slow parameter f/J depends ^{also on} time. Then we calculate Da, 14r 1

D[Pa,], ^{and the} simplified ^kinetic equation obtains the form

The equivalence of the Whitham method to the usual perturbation theory ^{based on two-}

scale representation has been checked for _many problems from non-linear wave theory. ^The generalization (6.4) ^was applied to autowave theory problems [19]. ^The application ^of (6.4) simplifies calculations appreciably. ^{This is} especially ^{true in a} multidimensional case (Sect. 7).

But in this item we describe on one-dimensional case. We should recall that in the one-

dimensional problem, ^{when T-} Ts, the free energy has the form

The dissipation has the form

Asymptotic substitution for p ^{leads to the} following ^formula

To calculate Das and 37,, ,, ^the following asymptotic ^{lemma is used.}

Lemma.

Denote 3,,

f££ ^{(p ( £x) einK’x, where we proceed with the} assumption ^{that the} function ço ^is

-c

smooth, ^{and 2 C} periodic, Ks ^{==" 0. Then for} any M > 0, when e is small enough

The lemma can be easily proved by ^{means of} integration by parts. ^{Now let us} substitute the form for _p and P into (6.5) ^and (6.6), respectively. ^We get

Then, according ^{to the} lemma, ^{all odd terms do not make a} contribution to !F as. ^{Let us} investigate the contribution of quadratic ^with respect ^{to P terms. Enter} f3 s

f3 (X s). ^Then

Since in the end we obtain

where

The coefficients a, 8, K ^are determined by ^formulas (4.2), and the coefficient is determined by

the formula

After the substitution of the obtained expressions (6.9), (6.10) ^into (6.4), taking ^the

variations with respect to f/J (* and gi *, ^{and the} replacement ^{of t with T} = e 2 t, ^{we obtain the}

Kuramoto-Tsuzuki equation (5.3).

7. Kinetic formalism in a général multidimensional case.

Formalism developed in section 6 is also true in a general case, if the substitution (6.2) ^is generalized properly. ^The substitution cited below is based on the alignment ^{of ideas} [ 17] ^with

the expression ^for p obtained for stable states [8].

The form of equation (3. 1) suggests the idea that it can be defined more exactly, ^{if we take}

for 37 with small e a more precise approximation than that of (2.8), (2.1). ^{If we} express 37 by p (x ), ^{the most} general expression for the free _energy with due regard ^for only ^{cubic terms and}

fourth _power terms has the form

Going ^{over to Fourier} representation, ^we get

Asymptotically precise expressions ^for Ê3, T4 ^{and the} expression ^for s (K ) ^{have been}

calculated [8]. ^It should be noted that F3, r4 depend only ^{on the} differences of

x-coordinates ^and quick-decreasing ^{functions of these} differences. If ^{we use} the simplified

formulas (2.8) ^and (2.1), ^{we obtain} approximations ⁱⁿ the form of 8-functions and their

derivatives for r 3 ^and r 4. Instead of the functions _s, r 3, r 4 ^the approximations appear

It will be quite clear from our later examinations that, in the limits .of the small values _E, the

error of these approximations ^is sufficiently small. The analysis of the kinetic equation (3.1)

will be carried out by ^means of applying 37 ^{as the} general expression (7.1 ). ^{Then the} generality

of formalism will become obvious, ^{and we can} easily observe the error that _may arise because of the approximation (7.3), i.e., ^{because of} applying (7.3) instead of precise ^{formulas. In}

contrast to the one-dimensional case, ^a multidimensional theory contains the whole sphere ^of

critical vectors : 1 K 1 = Ks. It results in the complication of asymptotic substitutions (6.7) ^and (5.1). ^{But one} should remember the formula applied by ^{Leibler to} the calculation of the free energy of different structures. Let Mn ^{be some fixed 2 n set} of critical modes. It means that

IKi 1 = ^{Ks is performed for all Ki e Mn (i}

^{1, 2,}

^{n). As Mn, one can choose, e.g.,}

Then the stable density Pas’ corresponding ^to Mn ^set takes the form

It should be noted that to make p real, ^{it is} necessary ^to satisfy ^the following condition : if K e Mn ^{then - K e} Mn ^{and the} complex conjugacy ^of corresponding qi. ^The generalization ^of (7.6) ^{to a} non-equilibrium ^{case is} sufficiently ^obvious. Then, ^{let us} suppose that

The Mn set, ^{with all} possible t, ^remains fixed. Let us dwell on the elucidation of the question

how does the Mn ^set arise in kinetics theory. ^{The form of this set} and its initial amplitudes

«Pn(X, r 1) ^are determined by initial conditions in the Cauchy initial values problem ^for (4.6).

Mn ^set is connected with the Fourier components of initial states, ^{which are not} damping ^at

the moment T 1. When t :> 7- 1, it is possible ^to simplify the kinetic equation (4.6), and reduce it to the system ^{of n} - equations ^{of a} reaction-diffusion form. To deduce them, ^{let us use the}

Whitham principle according ^to which the kinetic equation ^for amplitudes 4,j ^{must have a}

form

The calculation of :tas (see Appendix) ^{and of} Das results in

the sign ^{« + »} before £ 4 satisfies the condition x :::. X,, ^{and the « - »} sign, the condition

n

X X S. In this formula the Mn ^{set is fixed} by the initial condition p (K, 0 ) ^{and the} E3, E4 ^{sets of} triplets, ^{fours of} integral-value ^{indexes are} determined by ^the Mn ^{set. The} E3 ^set consists of the triplets of such numbers i l, i2, i3 ^{such that} Kil, Ki2, Ki, ^{E Mn and their}

total ^sum is equal ^{to zero ;} the definition of E4 ^is analogous.

Now we can answer the question ^{about the error} introduced by ^the approximation (2.8) ⁱⁿ

the free energy instead of the ^more precise ^free energy calculated in [8]. ^{First of} all, ^this approximation results in the exchange ^{of the statement}

for the expression ^K + 3 « . Then the functions r 3’ r 4 ^{are to be} exchanged ^{for the constants}

Ks

63, 64. ^{It is} possible since, ^{in formulas} (7.10), ail IKi [

Ks, ^{and so}

Therefore, ^when using approximations (2.8), (2.1) ⁱⁿ (7.10), Ê3, t 4 ^{should be} exchanged

for the expressions

Besides it should be noted that the dependence ^of G 3, Û4 ^{on po is} approximately ^{the same}

as that of F3 ^and F4. ^{This can} be determined by comparison ^{with the charts obtained} numerically ^for F3 ^and F4 ⁱⁿ [8]. Besides, the function F3 ^{does not} depend ^{on the} angles

contained by Kl l, Ki2’ Ki3’ ^if 1 Ki. 1, IKi21,

I K=3 I E ^{Mn and the function} F4 depends weakly.

Therefore, ^the approximation (7.11) turns out to be successful.

Finally, ^{the same} system ^of simplified ^kinetic equations ^{for the} amplitudes ^{takes the form}

When Mn

MI, ^{and there are} only ^{two vectors K and - K} in the collection of critical

modes, ^the equation system (7.12) ^{reduces to} Kuramoto-Tsuzuki equation ^{of a form (5.3).}

Despite the fact that the form of equations (7.12) ^is complicated, ^their general theory [16]

mentioned above allows us to obtain the description of the solutions’ behaviour. For « almost all » initial data f/Jj(X, T), ^{with t - oo,} the solutions tend ^to equilibrium ^{values of} f/Jj ^giving the stable minimum of the functional 5;-as 1 q’ 1, f/J 2, ..., f/J n] ] that follows from the

general theory ^{of the} equations ^{as such} [16]. ^{It should} be noted that such stable minima _may be several, particularly ^{if the} Mn ^{set contains many vectors. The time necessary for structure}

formation is estimated with respect ^{to formula} (5.9). The stable states will be put ^to analysis ⁱⁿ

the next section. It should be noted that, ⁱⁿ contrast to Leibler’s theory, ^{in our} approach ^all

main dependences XS, d, KS(N, a, ^p o) ^are obtained in analytical ^form.

8. Stable structures near Ts.

Let us examine the stables states that determine the microdomain structure. According ^to

5:F

section 7, they ^are given by Eiler-Lagrange equations & J’as & ipi (X, r)

^{0. To provide the stability}

of structures, 37as ^{must have a} local minimum. The structure form is determined by M,, ^set.

n

Since 3’ as contains the term a ^{1 V x 1/1} 1 2, a > 0, ^the necessary condition for 3’ as ^{minimum is}

i = 1

V xf/J i == ^{0. It means} that the stable configuration ^{of the} amplitude gi ^{does not} depend only ^on time, ^{but on} space variables ^as well. Therefore, ^a system ^of ordinary differential equations

can be applied ^to the numerical calculation of (Sect. 7)

where j

1, 2,

n, and:F as ^is :F as without the term with gradient.

It is known that the solution of the system ^as such, ^{with t} ---> oo, satisfies the conditions of minimum fas :

Therefore the stable amplitudes ipj ^{can be found by the} integration ^of (8.1). Besides, ^a

relaxation time can be estimated in the same way. It should be observed that, ^{if a linear}

dimension of the system ^is « E - 1, (8.1) ^{is a} system ^of asymptotically precise equations.

This system ^{can be} easily integrated numerically. ^{Such account} has been made for a

hexagonal configuration (n

3, M3 ^is determinated with respect ^to (7.5)), ^{and it shows that}

for a stable structure, ^{with t -} oo, all amplitudes 1 f/lj [ ^are equal (irrespective of initial data).

If the moduli 1 f/lj 1 ^{are equal then the} asymptotic formula for the free _energy of the stable

equilibrium configuration ^{is :}

where the designation ^{is entered is}

When using ^the simplified ^model (2.8), ^the coefficients Cn, dn ^are determined as follows :

Precise formulas (7.1) ^{lead to the same} expression (8.2), ^{but at the same} time, ^{in the}

formula for cn, G 3 ^{should be} exchanged ^for r3 and the formula for dn ^{assumes the form}

To plot ^a phase diagram ^{in the} simplified model, ^{let us use the facts} determined in [8].

1. The value _{X S} of the parameter y is smaller for these structures, ^{for which}

Ci 2 e2 ^is larger. Correspondingly Ts ^is smaller for the same structures.

n

2. When X - X S is negative and 1 X - X S 1 ^is increasing (i.e., y, X c and the increasing

of X ) ^{it is a structure with} smaller dn that has smaller free _energy.

3. The structures with larger n ^{are not} energetically favourable.

As it follows from these statements, ^a phase diagram determined by ^a comparison Y(") for different _n, when using ^this model, ^{does not} depend ^on numerical values of

G 3, G4, though X and ^a transition temperature Ts ^{of course do.}

Such a calculation programme ^was drawn _up in (7.10). ^Three configurations ^were put ^to Cn

analysis : ^lamellar (n

1 ), hexagonal (n

3 ) ^{and cubic} (n

6 ). The relations d n ^{for these} configurations ^{are connected} by ^the relationship :

Thus we came to the same phase diagram ^{as in} [8]. Using ^{our method we can} also obtain a

simple explicit expression ^for Xn. Here is the value of X parameter corresponding ^{to melt-}

structure transition temperature determined by Mn ^set (i.e., ^when n = 1, ^to lamellar;

n

3, ^to hexagonal, ^{and so} on).

Substituting ^the explicit expressions ^for G3, G 4, ^we get

Leibler showed that for _{p o} near 0.5, ^three types ^of structures could exist. They ^are transforming into each other due to temperature changes. ^But these considerations are not

supported by experiment. According ^{to it,} only ^{a lamellar} type ^structure arises. This effect

seems to be explained by the kinetic theory.

In fact the Mn ^{set for real initial} condition contains many ^{vectors K} with different

corresponding amplitudes .pK’ ^The analysis (not presented ^{in this} paper) ^of equation (8.1)

shows that at t > oo the only ^{one vector K} corresponds ^{to non-zero} amplitude. ^{This means}

that such a structure should be of lamellar type. ^This analysis has much in common with that

made by Haken in the Benard problem [20], ^{but is more} complicated. It will be submitted

elsewhere.

A stable final state should not only be determined by ^the global ^{minimum :F. It also} depends ^on initial conditions.

The study ^{of the} structures for _{p o} not close to 0.5 when _{X, c} is not small requires absolutely ^different methods, ^{since the} expansion ^into (4.1) series is incorrect in this region.

Actually, though ^the extrapolation of the results of this section leads to the right ^deductions that, when X » Xs, ^{there are lamellar} structures with the smallest free _energy, but it is

impossible ^to obtain the right expression ^{for the} period ^{of structure d. From the} study [6, 7]

based on scaling concepts, the formula d - aN 2/3 X ^1/6 follows, ^{while section 4} gives ^the dependence d - aN 1/2 . ^{The next section} investigates ^{a stable structure with} X > X S.

9. Stable structures at T Ts.

In this region ^{we will} investigate ^{stable states} only since, strictly speaking, the considerations about a dissipation function form are not applicable ^{in this case. Series} (4.1) ^{is also} unapplicable.

On the face of it, ^the investigation ^of Eiler-Lagrange equation ^is absolutely impossible

because of the complexity ^{of the} expression ^for f o and that of these high power (fourth) equations. But, just when X » X s, ^another asymptotic approach ^{based on the newest studies}

in the theory ^of solitary ^waves weak interaction is applicable [21]. ^{The main} ideas, ^{of which}

we give ^{a short account} only and without going ⁱⁿ many mathematical details, ^{are as follows.}

To make it simple, ^{let us} investigate the one-dimensional case to which lamellar structures

correspond, ^and put po

0.5. Then, ^{for an order} parameter ^{and free} energy density, get

The equilibrium equation ^{assumes the form}

It is common knowledge ^that structures, when X » y,, consist of domains divided by

narrow transition layers. ^Let the width of a domain be of d order, the width of a transition

layer ^{is order Li. Enter} a non-dimensional parameter v = 2013 . The asymptotic theory ^{is correct}

if v « 1 .

As calculations show, ^a limit v --* 0 corresponds ^{a limit} y, « x, i.e., ^{to the case of low} temperatures. ^{Let us} investigate ^{the structure of a solution as} such. When domain boundaries

are narrow, the plot ^{of P} (x) ^assumes the form of a saw-like line consisting ^of alternating ^line

segments ^with slopes + Î- ₄ and - 4 . ₄ It is connected with the fact that density ^deviation

p ^{in the domain is} approximately equal ^to ± 1 , ₂ excluding ^the transition layer. y ^{Let us denote}

the positions ^of intersection points of the saw-like line across x-axis : _q q « ... « q N. Then the transition layers have coordinates P1, P2,

^P N, where Pi = qi + 2 qi + i. ^{Let us suppose}

that there are neither narrow nor too wide domains, i.e.,

Then, ^{since we are} finding the solution minimizing f ^{_ , f’ dx, a} r ^{_Ç P 2 dx cannot be large.}

Then, taking into consideration the integral ^condition (2.1), and the form of the function

P (x), ^{we will see that the} component ^{with a P in} (9.3) ^{has a} power which is not larger ^than

the Const. a. d with all possible ^{x. In} the transition layer Px 1 , therefore, ^{the term}

d

,6Pxx ^is of 1 ^{power. If the} following inequality has been acted

then, ^when analysing-transition layer, ^{the term with a P can} be excluded from the equation,

since it is small in comparison ^with the others in the transition layer. Thus, first conditions

(9.4) and d/d ^{1 are} suggested ^and then, ^{when a} solution will be obtained, the observance of these conditions will be checked by substitution. Returning ^{to the} Eiler-Lagrange equation describing the i-transition layer, ^{it is} convenient to pass ^to variables p instead of

P, y

(x - Pi), where Pi ^{is a} position ^of i-layer localization point. Besides, ^we may put p(0) = ^{0. The} dependence J(y) for the transition layer ^can be obtained from the equation ^of

the 2nd _{power :}

This equation appears after the integration (9.3) ^{over y and} excluding ^{the term with}

a P . The constants of the integration (9.3) ^are chosen from the following condition : p - ^{0 should} be the solution of (9.5) ^{and be} equal ^{to zero.}

Equation (9.5) ^{is correct} only ^{in the} vicinity ^{of the} transition layer, i.e., ^when y - à. When 1 y 1 > 4, ^value p ^{must tend to the} constants p:, . Therefore, (9.5) ^{should be} completed by boundary conditions :

Equation (9.5) ^is already ^known [12]. ^{It can be} integrated standardly since it has energy

integral :

There is a nonperiodically (solitary) solution of Ppep for which lim p(y) ^{are constants}

l’ - ± 00

p + , being determined from the conditions :

that leads to approximate ^formulas

The physical meaning ^of (9.8) is that the boundary ^condition p == p -z. for domain bounds is - determined by ^the equilibrium ^{state of} homogeneous medium with the free _energy

fo(p).

When

here is a nonperiodic ^{solution that turns out to} be monotonic and satisfies (9.6). ^{A solution as}

such really describes the domain boundary. ^{Its form can} be calculated approximately, ^{if it is}

observed that in (9.5) ^{the term with} logarithms is much smaller (when y N » 1) ^{then the term}

with PX, ^if p ^{is not too close to} ± - . ^{Then it turns out that the} following ^formula may be ^a 2

good approximation

Then, ^if p per ^{is found, it is possible to calculate P (x) in a complete form. Actually, the sign}

selection in the domains p ... ± - _depends ^{on : which chains are} concentrated in domains A 2

and B. According g ^to (9.9), ( ) ^the p -density deviation from ± ₂ ^is exponentially ^{small and} may be neglected. ^As long ^as PX

p, the contribution of the term

a P 2 ^{dx is of integral nature.}

When integrating ^{this term} P may be considered to be the piece-linear ^{function mentioned}

above. By the condition (2.4) ^the integral ^{of P is} equal ^{to zero.} Then have :

M is domain numbers.

The complete ^free energy

where r ^{is the free} energy of a transition layer. ^{It can be} calculated with respect ^to (9.10) ^and (9.2) :

This formula can be obtained after substitution of (9.10) ^into (9.2), integrating ^with respect

to dy, ^and neglect ^{of the terms with} logarithms, owing ^to XN » ^{1. When the} layers ^are infinitely narrow, will _agree with precise value of its _energy.

We apply ^Whitham’s principle ^to establishing the coordinates of the transition layer (see

Sect. 6) :

since qj ^are parameters ⁱⁿ :F as’